Stability conditions of plastic deformation of thin-walled plates and shells under complex stress state

. Using examples of uniaxial and biaxial tension of thin-walled metal or composite elements, the main hypotheses allowing to estimate the conditions of unstable deformation are explained. Specific estimations are given for a basic equation relating stress and strain tensor intensities as applied to plates, cylindrical and spherical shells. The equipment for testing tubular specimens under proportional loading under various plane stress states is described.


Introduction
The ultimate stress-strain states of a material can be associated not only with loss of strength or with the onset of total yielding, but also with loss of deformation stability, for example, in uniaxial or biaxial tension of a plate or in the action of internal pressure in a cylindrical or spherical shell.The analysis of unstable deformation began with the modeling of the occurrence of necking in tensile cylindrical specimens, and it has a long history of interest to researchers from both theoretical [1] and applied [2][3][4][5][6] perspectives.The corresponding models describing the occurrence of unstable deformation are discussed in a number of monographs [7][8][9], and for more complex basic equation in [10][11].
The fundamental difference in the behavior of isotropic metals under uniaxial and biaxial tension was pointed out even by the S.P. Timoshenko -with references to the experimental work of earlier authors.Therefore, many works have been devoted to biaxial loading experiments, despite their complexity.Interesting results on twinning mechanisms during testing of thin-walled pipes in the fixture proposed in [12] were obtained in [13], and the interest in such tests has not waned to this day, although with more modern equipment [14].
The purpose of this review and methodological article was to briefly and accessibly explain and summarize the main hypotheses used in the analysis of such a specific and ambiguous phenomenon as the loss of deformation stability under a complex stress state.

Occurrence of deformation instability at uniaxial tension
Let us consider a simple model of self-developing deformation under uniaxial tension as an illustrative description of the occurrence of a neck in a flat or cylindrical specimen.Hereinafter, the "incompressibility hypothesis" (volume invariant under deformation) is adopted.For plastic deformation occurring mainly due to local shear, the hypothesis of volume invariant (v = 0.5) is fulfilled with sufficient accuracy.As longitudinal elongation proceeds, due to Poisson's effect the cross-sectional area decreases and the local "true" stress, referred to the current cross-sectional area, increases.At some point this process becomes unstable and the "neck", i.e. a zone of intense plastic deformation appears.
The model of self-developing deformation of a rod under a constant load P is as follows.Let us denote the initial and current lengths of the rod element L0, L, and the cross-sectional areas F0, F. The condition of volume invariant with application of linear deformation gives: Condition (1) with application of the logarithmic (natural) measure of deformation has the form: Expressing the conditional stress in terms of the initial section: 0 0 , P F  the true stress e P F  we obtain from (1): (3) i.e. the true stress (per decreasing current cross-sectional area) increases with increasing longitudinal strain.
The basic equation (elastic-plastic deformation diagram) for true stresses can be written as: where    is a monotonically increasing nonlinear function.In the experimental tensile diagram 0    (Fig. 1, a), the conditional stress is usually plotted: and after reaching the critical strain *  of deformation stability loss begins to decrease, and a falling branch of the diagram appears (Fig. 1, a).The condition for the appearance of instability is the equality to zero of the strain derivative of the conditional stress (5):

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It is clear from ( 6) that the derivative   Examples.Let us take (4) as the basic equation Substituting ( 7) into condition (6), we find: For the logarithmic strain (2), adopting by analogy with (7) the basic equation approximation of the true diagram, we obtain an expression similar to (8): The critical strain in this simplified formulation depends only on the nonlinearity index of deformation.At linear elastic behavior (m=1), the effect of instability of deformation, i.e., necking, according to (8) does not manifest itself.Such a categorical conclusion is related to the simplicity of the considered model, with the assumption of homogeneous deformation.Non-uniformity of deformation also occurs in elastic deformation due to various structural and geometric inhomogeneities.In addition, it is not necessary to consider the Poisson's ratio ν equal to 0.5 (the condition of volume invariant).The effect of true stress growth at crosssectional reduction as a result of deformation is always manifested, unless it is an exotic material like cork with v = 0 or a structurally inhomogeneous auxetics with a negative Poisson's ratio.
To analyze unstable deformation, more complicated problems were solved taking into account self-heating during plastic deformation and with the criterion of conservation of the material mass rather than its volume.If the bulk modulus of elasticity is not zero (v ≠ 0.5), the deformation results in a change in density (proportional to the change in volume), but the mass of a given element of the sample is conserved, which can be accounted for in a conservation equation such as (1).

Hypotheses of the model of deformation of incompressible material under proportional biaxial tension
The stress-strain state in an isotropic material can be described by three principal normal stresses 1 2 3 ; ; ,    acting at mutually perpendicular sites and three principal strains ;;    .
When analyzing the conditions for the onset of unstable deformation, a number of assumptions are traditionally made.
1.The plane stress state realized in thin plates and thin-walled shells means: 3 0  .
2. The condition of constant volume V corresponds to the value of Poisson's ratio 0.5 v  .Indeed, from Hooke's law: 3. Under the complex stress state, there is a single relation of the type (4), (7), ( 9) between the intensities of stress tensor i  and strain tensor i  : 4. The deformation intensity is expressed in the general case and taking into account incompressibility ( Under uniaxial tension from (12): For plane deformation: With uniaxial tension (α=0), of course: . From Hooke's law (10) under proportional loading: The loading is assumed to be simple (proportional), which means that the ratios of the principal strains to the corresponding components of the stress deviator are equal, since (following Mises and Genki) the globular part σ of the stress tensor is assumed to have no effect on the ultimate stress: It is now possible to express the principal strains through the intensity:

Occurrence of deformation instability in biaxial tension
What is the condition of instability in biaxial tension?Fig. 1 shows that if the stress and strain intensities are plotted along the axes, then, as in the uniaxial case, the critical strain intensity corresponds to the equality of the derivative (of the stress intensity on the strain intensity) to the ratio of the stress intensity to the size of the so-called "subtangent" z.It follows from Drucker's postulate [5] that the deformation is stable as long as the work of increments of applied forces on virtual increments of displacements is positive, whence, as will be shown below, follows (17): Note that the criterion for the appearance of deformation instability is only one of a number of other limit state criteria, including strength, plasticity and energy criteria for crack growth.

Criteria for the occurrence of unstable deformation at different ratios between principal stresses
Consider an element (Fig. 2) with dimensions x and y in the plane 1-2 and thickness h.Along axes 1 and 2, forces P1 and P2 are applied to the element, through which the principal stresses σ1, σ2 are expressed: 1 1 2 2 ;.

P yh P xh  
  The forces P1, P2, on the one hand, increase due to plastic hardening according to the generalized diagram (11), and on the other hand, decrease due to a decrease in the area of the corresponding stressed areas.Stable deformation corresponds to the possibility of growth of forces Pi: dPi>0.Unstable deformation occurs when the differential of one of the forces turns to zero dPi = 0.
1) When 12 >  the instability condition has the form: To find the critical strain, we need to know the generalized nonlinear deformation diagram (11), i.e., the basic equation relating stress and strain intensities, which for simplicity is assumed to be in form: 2) When 12    the condition of instability has the form: Estimates of critical strains are important in the development of technologies for producing thin-walled metal parts by rolling and deep drawing methods [2][3][4].
3) In general, the condition of instability (17) according to Drucker's postulate [5] corresponds to the equality to zero of the sum of work of increments of forces dP1 and dP2 on virtual displacements dx and dy: We use the basic nonlinear equation for intensities in the simplest basic equation form: We have obtained a match for logarithmic strains with uniaxial tension.Now we can investigate the dependence of ε* on α.The basic equation ( 7), ( 22) has a fundamental drawback -the impossibility to determine the (Young's) modulus at zero (infinitely small) deformation: But in the linear case (n = 1) all considerations about loss of stability, about the length of the subtangent z, about finding the critical deformation lose their meaning.
This condition (23) of smoothness of the diagram at the point of transition from the linear to the nonlinear section is not necessary, and the discontinuity of the derivative at the end of the linear section will not affect the analysis of deformation instability, since the linear section (and Young's modulus) is not taken into account at all in the analysis of instability.But if we consider the elastic limit negligibly small compared to the further nonlinear deformation, we can use condition (23), which was applied by Swift [1] and Malinin [5].In general, the constitutive relation (23) can reflect the presence of initial stresses at zero strains, which makes it possible to correctly relate the nonlinear diagram to the Young's modulus, in contrast to the basic equation ( 22).
The dependences ( 19)-( 21) of the critical strain on α are shown in Fig. 3.Note that the dependence according to the curve z1-1, plotted by formula (19), 2 is valid only at α<1, since σ1 ˃ σ2 was assumed in its derivation.Accordingly, the curve (20) -z2-2 is valid only at α>1.And the dependence (21) -z3 -3 is derived from a more general stability criterion and it is valid over the entire range of α, showing qualitative agreement with dependences (19)-1 and (20)-2 in the ranges of their validity.
Let us perform the simplest analysis of the extrema of the dependence (21) of z on α:

Estimates of the conditions of unstable deformation in thinwalled cylindrical and spherical shells
For ductile materials, a single diagram under proportional loading can be approximated by a basic equation ( 9) using a logarithmic strain measure.
When the tube is stretched in the circumferential direction by internal pressure p, the tube element can be represented as a ring with a square cross-section, wall thickness h and width h.The volume of such a thin-walled ring is  This dependence (25) p(R) has a maximum, which corresponds to the critical stress and strain: The critical strain is found to be significantly lower than that of uniaxial longitudinal tensile strain.
3. Under the action of internal pressure p and by a longitudinal force P on a thin-walled tube with radius R and wall thickness h, let us assume (see Fig. 2) the longitudinal coordinate (x is the length of the element along axis 1) for axis 1 and the circumferential coordinate (y is the size of the element in the direction of axis 2) for axis 2. The condition (21) for the occurrence of deformation instability has the form: E3S Web of Conferences 458, 07010 (2023) EMMFT-2023 https://doi.org/10.1051/e3sconf/202345807010 Now let us express stresses and strains through intensities and use the condition ( 16) of loading proportionality and equality of strain ratios to the corresponding components of stress deviators: The lower formula (28) is plotted in Fig. 3 -curve 4, from which it can be seen that the critical strain for the tube is much lower than for the plate (curve 3), which is also evident when comparing the tube stretching with the internal pressure.This qualitatively explains the experiments described by S.P. Timoshenko on annealed isotropic copper samples.Tubular samples sealed at the ends failed at a strain of 2-3%, and flat tensile samples -at a strain of 10-20%.
Let us analyze, as above, the extrema of the relation (28) of z on α:  An interesting result is obtained -the possibility of determining the critical ratio between the stresses that provide the minimum and maximum ultimate strain in biaxial tension (see Fig. 3).
6.4.For a spherical shell with radius R and wall thickness h under the action of internal pressure p, unstable deformation occurs under the following condition:   In biaxial tension, a higher necessary ultimate strain at the onset of unstable deformation is predicted, but it should be remembered that in biaxial, and especially in triaxial tension, the occurrence of plastic deformations (e.g., near the crack tip) is hindered, which leads to a decrease in fracture toughness and to an increase in the probability of failure by another, more dangerous mechanism -in the form of brittle crack growth rather than in the form of unstable deformation.
7 Methods of testing tubular specimens for complex stress state 1.Biaxial loading of flat specimens.Testing thin-walled tubular specimens under complex stress state conditions is simpler, but the composite does not exist outside the constructure, and a composite plate cannot be made into a pipe, just as a pipe cannot be made into a plate [9].Therefore, it is necessary to directly test flat elements.For this purpose, in particular, the design of a cross specimen has been developed, the working zone of which is located in its middle, and different loads are applied to the protrusions (Fig. 4).Biaxial loading systems can be: 1) independent (two rupture testing machines connected at right angles), 2) coupled (two hydraulic cylinders from a single pressure source), or 3) kinematically coupled (via a system of straps, cables, rollers).Note that tests under proportional loading, where each component of the stress tensor varies in proportion to a single parameter (e.g., time, or oil pressure in the loading system), are considered standard, comparable tests.Using special fixtures, it is possible to create also shear stresses in cross specimens [14], which at the same time allow to realize the general case of plane stress state.2. Testing of tubular specimens in tension, compression, torsion, internal and external pressure.Testing of thin-walled tubular specimens is of interest for a number of reasons [15,16]: -they reflect the technology of obtaining and properties of winding composite products; -practically homogeneous (over a small wall thickness) stress state is realized in them; -it is easy to realize a plane stress state of the general type (biaxial tension/compression and torsion/shear).
The latter is particularly important: it is easy to apply tensile (compressive) axial force P, torque M, internal (external) pressure p to tubular specimens, resulting in all three components of the stress tensor for the plane stress state:  3. Fixture for obtaining a proportional biaxial stress state in tubes.An original method of creating a biaxial stress state in tubes under the action of internal pressure alone was proposed in [12], and it consists in the application of special terminations in the setup, the scheme of which is shown in Fig. 5.
Indeed, if the end of the tube is plugged, the traditional, simplified expressions for the axial σz and circumferential σθ stresses have, respectively, the following form: where p is the internal pressure, R, h are the radius and wall thickness of the tube.If a solid cylinder of radius R is introduced inside the end part of the tube on a sliding fit, then the axial force will not act on the tube (σz = 0) and a purely circumferential tension will result.
If the end of the tube is plugged, but a hole is left in the end with a slip-fit cylinder of radius r inserted, the axial stress will be reduced compared to (31): by another simple transformation of the termination with the introduction of a slip-fit ring on the outside of the termination, compressive axial stresses proportional to the internal pressure and circumferential stresses can also be created in the specimen.
Numerous careful experiments on thin-walled tubular steel and titanium specimens [13] in the setup shown in Fig. 5, confirmed the occurrence of different failure mechanisms (twinning, brittle cracks, deformation instability) depending on the type of plane stress state.

Conclusions
Traditional hypotheses: incompressibility, loading proportionality, presence of a single nonlinear deformation diagram allow, on the basis of the Drucker-type stability postulate, to estimate the condition for the occurrence of self-developing plastic deformation, which depends on the ratio of the two principal stresses.This condition be checked when assessing critical states of structural elements along with other conditions: stresses reaching the strength or plasticity limit, fulfillment of the criterion of crack initiation and growth within the framework of linear or nonlinear fracture mechanics.


at the point corresponding to the maximum of the conditional diagram is equal to the value of the function    related to   1 .The graphical illustration of formula (6) is shown in Fig. 1, b: a segment of unit length is drawn to the left of the origin along the ε-axis and a tangent is drawn from its end to   e     .The point of tangency defines the critical strain *  .

Fig. 2 .
Fig. 2. Schematic diagram of the element with dimensions and applied forces

2 2
Rh , and from the condition of volume invariant in the process of circumferential stretching, i.e. at growth of radius R, we can write the and initial radius, wall thickness and internal pressure, respectively.

Fig. 4 .
Fig. 4. Schematic of a cross flat specimen for testing under complex stress state where θ, z are the circumferential and longitudinal coordinates (Fig.5).